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Theorem prdssca 16724
 Description: Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (𝜑𝑆𝑉)
prdsbas.r (𝜑𝑅𝑊)
Assertion
Ref Expression
prdssca (𝜑𝑆 = (Scalar‘𝑃))

Proof of Theorem prdssca
Dummy variables 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . . 4 𝑃 = (𝑆Xs𝑅)
2 eqid 2801 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3 eqidd 2802 . . . 4 (𝜑 → dom 𝑅 = dom 𝑅)
4 eqidd 2802 . . . 4 (𝜑X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)))
5 eqidd 2802 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
6 eqidd 2802 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
7 eqidd 2802 . . . 4 (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
8 eqidd 2802 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
9 eqidd 2802 . . . 4 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10 eqidd 2802 . . . 4 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
11 eqidd 2802 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
12 eqidd 2802 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
13 eqidd 2802 . . . 4 (𝜑 → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
14 prdsbas.s . . . 4 (𝜑𝑆𝑉)
15 prdsbas.r . . . 4 (𝜑𝑅𝑊)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15prdsval 16723 . . 3 (𝜑𝑃 = (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
17 eqid 2801 . . 3 (Scalar‘𝑃) = (Scalar‘𝑃)
18 scaid 16628 . . 3 Scalar = Slot (Scalar‘ndx)
1914elexd 3464 . . 3 (𝜑𝑆 ∈ V)
20 snsstp1 4712 . . . . 5 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}
21 ssun2 4103 . . . . 5 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
2220, 21sstri 3927 . . . 4 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
23 ssun1 4102 . . . 4 ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
2422, 23sstri 3927 . . 3 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
2516, 17, 18, 19, 24prdsvallem 16722 . 2 (𝜑 → (Scalar‘𝑃) = 𝑆)
2625eqcomd 2807 1 (𝜑𝑆 = (Scalar‘𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ∪ cun 3882   ⊆ wss 3884  {csn 4528  {cpr 4530  {ctp 4532  ⟨cop 4534   class class class wbr 5033  {copab 5095   ↦ cmpt 5113   × cxp 5521  dom cdm 5523  ran crn 5524   ∘ ccom 5527  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  1st c1st 7673  2nd c2nd 7674  Xcixp 8448  supcsup 8892  0cc0 10530  ℝ*cxr 10667   < clt 10668  ndxcnx 16475  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  ·𝑖cip 16565  TopSetcts 16566  lecple 16567  distcds 16569  Hom chom 16571  compcco 16572  TopOpenctopn 16690  ∏tcpt 16707   Σg cgsu 16709  Xscprds 16714 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-hom 16584  df-cco 16585  df-prds 16716 This theorem is referenced by:  pwssca  16764  xpssca  16844  xpsvsca  16845  prdslmodd  19737  dsmmlss  20436  rrxsca  24003
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